As with any kind of examination, this can vary depending on the kind and level. Philosophy courses (undergraduate or graduate) are specific to an era, school, region, or even down to particular works: e.g., "Modern European Social Theory", "20th Century Empiricism", "The Existentialist Movement", "Hegel's Phenomenology of Spirit", "Philosophy of Ancient Greece", etc. Examinations in such courses are designed to test mastery of the specific ideas within those specific topic-areas, as well as basic critical thinking skills, like understanding the evolution of ideas within those topic areas or grasping the nuanced differences between different authors within the same basic worldview.
Graduate work in philosophy is usually a deep dive into a particular subfield. Examinations on that level are meant to show that a candidate can craft and defend a philosophical argument that stands up against professional critique. That's no different than what happens to a candidate in physics or mathematics; the metrics of critique may differ, but the process is the same. Philosophy may look 'unstable' to someone with little grounding in the material, but that's only because people without grounding in the material do not see the rational structures that underly and constrain the arguments. Remember, to someone who is naive to physics, the sun appears to rise in the East and set in the West. Once one understands some foundational principles in physics, that obvious perception is precluded, but to (for instance) a Flat-Earther the idea that the sun physically rises and sets is perfectly feasible.
All of modern academia, without exception, is based in the traditional Master/Journeyman relationship of the early university system (before all of our modern disciplines split away from philosophy general). In a Master/Journeyman relationship, the goal is to ensure that the journeyman is competent enough in the material that s'he will not embarrass him/herself, the master, other members of the discipline, or the discipline itself. Examinations thus serve two purposes:
- To guarantee the journeyman is qualified and prepared to do the work on the level of a master
- To mark the passage from journeyman status to master status
Examinations don't need to have a purpose beyond that, because a final examination merely marks the beginning of a professional career.